The answer to 'why is my supply line slow draining' is that the supply line could be clogged or restricted. This is often caused by mineral buildup, debris, or a kink in the line. Symptoms include slow water flow, reduced water pressure, and the supply line taking longer to fill up.
To fix a slow-draining supply line, you can try cleaning the line by flushing it with vinegar or a commercial line cleaner. If that doesn't work, you may need to replace the supply line. If you're not comfortable doing this yourself, it's best to call a professional plumber to inspect and repair the issue.